\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  21.0 & -3.00 x_{1} & -6.00 x_{2} & -7.00 x_{3} & -3.00 x_{4} & +  9.00 x_{5} & -1.00 x_{6}\\
 x_{8}   &  40.0 & +  1.00 x_{1} & -8.00 x_{2} & +  1.00 x_{3} & -10.00 x_{4} & -2.00 x_{5} & +  8.00 x_{6}\\
 x_{9}   &  71.0 & -5.00 x_{1} & -7.00 x_{2} & -9.00 x_{3} & -7.00 x_{4} & +  1.00 x_{5} & -9.00 x_{6}\\
 x_{10}   &  -4.0 & +  7.00 x_{1} & -8.00 x_{2} & +  1.00 x_{3} & +  3.00 x_{4} & + 10.00 x_{5} & +  7.00 x_{6}\\
 x_{11}   &  -2.0 & -4.00 x_{1} & +  2.00 x_{2} & -10.00 x_{3} & -2.00 x_{4} & + 10.00 x_{5} & + 10.00 x_{6}\\
 x_{12}   &  66.0 & -8.00 x_{1} & -9.00 x_{2} & -5.00 x_{3} &    &   & -10.00 x_{6}\\
 x_{13}   &  -21.0 & +  7.00 x_{1} & -9.00 x_{2} & + 10.00 x_{3} & +  4.00 x_{4} & +  9.00 x_{5} & +  8.00 x_{6}\\
\hline
z    &  0.0 & -3.00 x_{1} & +  3.00 x_{2} & -2.00 x_{3} & +  1.00 x_{4} & -2.00 x_{5} & -2.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  3.00 y_{7} & -1.00 y_{8} & +  5.00 y_{9} & -7.00 y_{10} & +  4.00 y_{11} & +  8.00 y_{12} & -7.00 y_{13}\\
 y_{2}   &  1.0 & +  6.00 y_{7} & +  8.00 y_{8} & +  7.00 y_{9} & +  8.00 y_{10} & -2.00 y_{11} & +  9.00 y_{12} & +  9.00 y_{13}\\
 y_{3}   &  1.0 & +  7.00 y_{7} & -1.00 y_{8} & +  9.00 y_{9} & -1.00 y_{10} & + 10.00 y_{11} & +  5.00 y_{12} & -10.00 y_{13}\\
 y_{4}   &  1.0 & +  3.00 y_{7} & + 10.00 y_{8} & +  7.00 y_{9} & -3.00 y_{10} & +  2.00 y_{11} &   & -4.00 y_{13}\\
 y_{5}   &  1.0 & -9.00 y_{7} & +  2.00 y_{8} & -1.00 y_{9} & -10.00 y_{10} & -10.00 y_{11} &   & -9.00 y_{13}\\
 y_{6}   &  1.0 & +  1.00 y_{7} & -8.00 y_{8} & +  9.00 y_{9} & -7.00 y_{10} & -10.00 y_{11} & + 10.00 y_{12} & -8.00 y_{13}\\
\hline
z    &  -0 & -21.00 y_{7} & -40.00 y_{8} & -71.00 y_{9} & +  4.00 y_{10} & +  2.00 y_{11} & -66.00 y_{12} & + 21.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.284210526316 & +  0.52 y_{7} & -0.76 y_{8} & +  0.14 y_{9} & +  0.16 y_{5} & -4.86 y_{10} & +  5.21 y_{12} & +  0.56 y_{3}\\
 y_{2}   &  1.93684210526 & +  6.66 y_{7} & +  8.17 y_{8} & + 11.75 y_{9} & -0.37 y_{5} & +  3.75 y_{10} & + 11.84 y_{12} & -0.57 y_{3}\\
 y_{13}   &  0.105263157895 & -0.11 y_{7} & +  0.05 y_{8} & +  0.42 y_{9} & -0.05 y_{5} & -0.58 y_{10} & +  0.26 y_{12} & -0.05 y_{3}\\
 y_{4}   &  0.589473684211 & +  1.81 y_{7} & + 10.09 y_{8} & +  4.36 y_{9} & +  0.11 y_{5} & -1.64 y_{10} & -1.53 y_{12} & +  0.31 y_{3}\\
 y_{11}   &  0.00526315789474 & -0.81 y_{7} & +  0.15 y_{8} & -0.48 y_{9} & -0.05 y_{5} & -0.48 y_{10} & -0.24 y_{12} & +  0.05 y_{3}\\
 y_{6}   &  0.105263157895 & +  9.89 y_{7} & -9.95 y_{8} & + 10.42 y_{9} & +  0.95 y_{5} & +  2.42 y_{10} & + 10.26 y_{12} & -0.05 y_{3}\\
\hline
z    &  2.22105263158 & -24.82 y_{7} & -38.59 y_{8} & -63.12 y_{9} & -1.21 y_{5} & -9.12 y_{10} & -60.95 y_{12} & -1.01 y_{3}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  24.8210526316 & -0.52 x_{1} & -6.66 x_{2} & +  0.11 x_{13} & -1.81 x_{4} & +  0.81 x_{11} & -9.89 x_{6}\\
 x_{8}   &  38.5894736842 & +  0.76 x_{1} & -8.17 x_{2} & -0.05 x_{13} & -10.09 x_{4} & -0.15 x_{11} & +  9.95 x_{6}\\
 x_{9}   &  63.1157894737 & -0.14 x_{1} & -11.75 x_{2} & -0.42 x_{13} & -4.36 x_{4} & +  0.48 x_{11} & -10.42 x_{6}\\
 x_{5}   &  1.21052631579 & -0.16 x_{1} & +  0.37 x_{2} & +  0.05 x_{13} & -0.11 x_{4} & +  0.05 x_{11} & -0.95 x_{6}\\
 x_{10}   &  9.11578947368 & +  4.86 x_{1} & -3.75 x_{2} & +  0.58 x_{13} & +  1.64 x_{4} & +  0.48 x_{11} & -2.42 x_{6}\\
 x_{12}   &  60.9473684211 & -5.21 x_{1} & -11.84 x_{2} & -0.26 x_{13} & +  1.53 x_{4} & +  0.24 x_{11} & -10.26 x_{6}\\
 x_{3}   &  1.01052631579 & -0.56 x_{1} & +  0.57 x_{2} & +  0.05 x_{13} & -0.31 x_{4} & -0.05 x_{11} & +  0.05 x_{6}\\
\hline
z    &  -2.22105263158 & -0.28 x_{1} & -1.94 x_{2} & -0.11 x_{13} & -0.59 x_{4} & -0.01 x_{11} & -0.11 x_{6}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  24.8210526316 & -0.52 x_{1} & -6.66 x_{2} & +  0.11 x_{13} & -1.81 x_{4} & +  0.81 x_{11} & -9.89 x_{6}\\
 x_{8}   &  38.5894736842 & +  0.76 x_{1} & -8.17 x_{2} & -0.05 x_{13} & -10.09 x_{4} & -0.15 x_{11} & +  9.95 x_{6}\\
 x_{9}   &  63.1157894737 & -0.14 x_{1} & -11.75 x_{2} & -0.42 x_{13} & -4.36 x_{4} & +  0.48 x_{11} & -10.42 x_{6}\\
 x_{5}   &  1.21052631579 & -0.16 x_{1} & +  0.37 x_{2} & +  0.05 x_{13} & -0.11 x_{4} & +  0.05 x_{11} & -0.95 x_{6}\\
 x_{10}   &  9.11578947368 & +  4.86 x_{1} & -3.75 x_{2} & +  0.58 x_{13} & +  1.64 x_{4} & +  0.48 x_{11} & -2.42 x_{6}\\
 x_{12}   &  60.9473684211 & -5.21 x_{1} & -11.84 x_{2} & -0.26 x_{13} & +  1.53 x_{4} & +  0.24 x_{11} & -10.26 x_{6}\\
 x_{3}   &  1.01052631579 & -0.56 x_{1} & +  0.57 x_{2} & +  0.05 x_{13} & -0.31 x_{4} & -0.05 x_{11} & +  0.05 x_{6}\\
\hline
z    &  -4.44210526316 & -1.57 x_{1} & +  1.13 x_{2} & -0.21 x_{13} & +  1.82 x_{4} & -0.01 x_{11} & -0.21 x_{6}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  24.8210526316 & -0.52 x_{1} & -6.66 x_{2} & +  0.11 x_{13} & -1.81 x_{4} & +  0.81 x_{11} & -9.89 x_{6}\\
 x_{8}   &  38.5894736842 & +  0.76 x_{1} & -8.17 x_{2} & -0.05 x_{13} & -10.09 x_{4} & -0.15 x_{11} & +  9.95 x_{6}\\
 x_{9}   &  63.1157894737 & -0.14 x_{1} & -11.75 x_{2} & -0.42 x_{13} & -4.36 x_{4} & +  0.48 x_{11} & -10.42 x_{6}\\
 x_{5}   &  1.21052631579 & -0.16 x_{1} & +  0.37 x_{2} & +  0.05 x_{13} & -0.11 x_{4} & +  0.05 x_{11} & -0.95 x_{6}\\
 x_{10}   &  9.11578947368 & +  4.86 x_{1} & -3.75 x_{2} & +  0.58 x_{13} & +  1.64 x_{4} & +  0.48 x_{11} & -2.42 x_{6}\\
 x_{12}   &  60.9473684211 & -5.21 x_{1} & -11.84 x_{2} & -0.26 x_{13} & +  1.53 x_{4} & +  0.24 x_{11} & -10.26 x_{6}\\
 x_{3}   &  1.01052631579 & -0.56 x_{1} & +  0.57 x_{2} & +  0.05 x_{13} & -0.31 x_{4} & -0.05 x_{11} & +  0.05 x_{6}\\
\hline
z    &  -4.44210526316 & -1.57 x_{1} & +  1.13 x_{2} & -0.21 x_{13} & +  1.82 x_{4} & -0.01 x_{11} & -0.21 x_{6}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  8.61235955056 & -9.16 x_{1} & +  1.78 x_{10} & -0.92 x_{13} & -4.73 x_{4} & -0.05 x_{11} & -5.59 x_{6}\\
 x_{8}   &  18.7191011236 & -9.84 x_{1} & +  2.18 x_{10} & -1.31 x_{13} & -13.67 x_{4} & -1.20 x_{11} & + 15.22 x_{6}\\
 x_{9}   &  34.5393258427 & -15.38 x_{1} & +  3.13 x_{10} & -2.24 x_{13} & -9.51 x_{4} & -1.02 x_{11} & -2.83 x_{6}\\
 x_{5}   &  2.10674157303 & +  0.32 x_{1} & -0.10 x_{10} & +  0.11 x_{13} & +  0.06 x_{4} & +  0.10 x_{11} & -1.19 x_{6}\\
 x_{2}   &  2.43258426966 & +  1.30 x_{1} & -0.27 x_{10} & +  0.15 x_{13} & +  0.44 x_{4} & +  0.13 x_{11} & -0.65 x_{6}\\
 x_{12}   &  32.1404494382 & -20.58 x_{1} & +  3.16 x_{10} & -2.09 x_{13} & -3.66 x_{4} & -1.28 x_{11} & -2.61 x_{6}\\
 x_{3}   &  2.39325842697 & +  0.18 x_{1} & -0.15 x_{10} & +  0.14 x_{13} & -0.06 x_{4} & +  0.03 x_{11} & -0.31 x_{6}\\
\hline
z    &  -1.70224719101 & -0.11 x_{1} & -0.30 x_{10} & -0.04 x_{13} & +  2.31 x_{4} & +  0.13 x_{11} & -0.94 x_{6}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.13681183237 & -5.76 x_{1} & +  1.02 x_{10} & -0.47 x_{13} & +  0.35 x_{8} & +  0.37 x_{11} & -10.86 x_{6}\\
 x_{4}   &  1.36894001643 & -0.72 x_{1} & +  0.16 x_{10} & -0.10 x_{13} & -0.07 x_{8} & -0.09 x_{11} & +  1.11 x_{6}\\
 x_{9}   &  21.5267050123 & -8.54 x_{1} & +  1.62 x_{10} & -1.32 x_{13} & +  0.70 x_{8} & -0.19 x_{11} & -13.41 x_{6}\\
 x_{5}   &  2.18364831553 & +  0.28 x_{1} & -0.09 x_{10} & +  0.10 x_{13} & -0.00 x_{8} & +  0.09 x_{11} & -1.12 x_{6}\\
 x_{2}   &  3.03245686113 & +  0.98 x_{1} & -0.20 x_{10} & +  0.11 x_{13} & -0.03 x_{8} & +  0.09 x_{11} & -0.16 x_{6}\\
 x_{12}   &  27.1261298274 & -17.94 x_{1} & +  2.58 x_{10} & -1.74 x_{13} & +  0.27 x_{8} & -0.96 x_{11} & -6.69 x_{6}\\
 x_{3}   &  2.31635168447 & +  0.22 x_{1} & -0.16 x_{10} & +  0.15 x_{13} & +  0.00 x_{8} & +  0.03 x_{11} & -0.38 x_{6}\\
\hline
z    &  1.46631059984 & -1.77 x_{1} & +  0.07 x_{10} & -0.26 x_{13} & -0.17 x_{8} & -0.07 x_{11} & +  1.64 x_{6}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.196821192053 & -0.53 x_{1} & +  0.09 x_{10} & -0.04 x_{13} & +  0.03 x_{8} & +  0.03 x_{11} & -0.09 x_{7}\\
 x_{4}   &  1.5880794702 & -1.31 x_{1} & +  0.26 x_{10} & -0.14 x_{13} & -0.04 x_{8} & -0.05 x_{11} & -0.10 x_{7}\\
 x_{9}   &  18.8863576159 & -1.42 x_{1} & +  0.35 x_{10} & -0.74 x_{13} & +  0.27 x_{8} & -0.64 x_{11} & +  1.24 x_{7}\\
 x_{5}   &  1.96264900662 & +  0.88 x_{1} & -0.20 x_{10} & +  0.15 x_{13} & -0.04 x_{8} & +  0.06 x_{11} & +  0.10 x_{7}\\
 x_{2}   &  3.00132450331 & +  1.07 x_{1} & -0.21 x_{10} & +  0.12 x_{13} & -0.04 x_{8} & +  0.08 x_{11} & +  0.01 x_{7}\\
 x_{12}   &  25.8092715232 & -14.39 x_{1} & +  1.95 x_{10} & -1.45 x_{13} & +  0.05 x_{8} & -1.18 x_{11} & +  0.62 x_{7}\\
 x_{3}   &  2.2421192053 & +  0.42 x_{1} & -0.20 x_{10} & +  0.16 x_{13} & -0.01 x_{8} & +  0.02 x_{11} & +  0.03 x_{7}\\
\hline
z    &  1.78887417219 & -2.64 x_{1} & +  0.22 x_{10} & -0.33 x_{13} & -0.12 x_{8} & -0.01 x_{11} & -0.15 x_{7}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  1.14486434109 & -0.11 x_{1} & -0.48 x_{5} & +  0.03 x_{13} & +  0.01 x_{8} & +  0.06 x_{11} & -0.04 x_{7}\\
 x_{4}   &  4.24583333333 & -0.12 x_{1} & -1.35 x_{5} & +  0.06 x_{13} & -0.09 x_{8} & +  0.03 x_{11} & +  0.04 x_{7}\\
 x_{9}   &  22.4465116279 & +  0.16 x_{1} & -1.81 x_{5} & -0.47 x_{13} & +  0.20 x_{8} & -0.54 x_{11} & +  1.42 x_{7}\\
 x_{10}   &  10.0509689922 & +  4.48 x_{1} & -5.12 x_{5} & +  0.78 x_{13} & -0.20 x_{8} & +  0.29 x_{11} & +  0.53 x_{7}\\
 x_{2}   &  0.871317829457 & +  0.12 x_{1} & +  1.09 x_{5} & -0.05 x_{13} & +  0.01 x_{8} & +  0.02 x_{11} & -0.10 x_{7}\\
 x_{12}   &  45.3596899225 & -5.67 x_{1} & -9.96 x_{5} & +  0.07 x_{13} & -0.34 x_{8} & -0.62 x_{11} & +  1.65 x_{7}\\
 x_{3}   &  0.26996124031 & -0.46 x_{1} & +  1.00 x_{5} & +  0.01 x_{13} & +  0.03 x_{8} & -0.04 x_{11} & -0.07 x_{7}\\
\hline
z    &  4.03013565891 & -1.64 x_{1} & -1.14 x_{5} & -0.16 x_{13} & -0.16 x_{8} & +  0.05 x_{11} & -0.03 x_{7}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  1.56234718826 & -0.82 x_{1} & +  1.07 x_{5} & +  0.04 x_{13} & +  0.06 x_{8} & -1.55 x_{3} & -0.15 x_{7}\\
 x_{4}   &  4.43031784841 & -0.44 x_{1} & -0.67 x_{5} & +  0.07 x_{13} & -0.07 x_{8} & -0.68 x_{3} & -0.01 x_{7}\\
 x_{9}   &  18.7555012225 & +  6.44 x_{1} & -15.55 x_{5} & -0.58 x_{13} & -0.24 x_{8} & + 13.67 x_{3} & +  2.37 x_{7}\\
 x_{10}   &  12.0317848411 & +  1.11 x_{1} & +  2.25 x_{5} & +  0.85 x_{13} & +  0.03 x_{8} & -7.34 x_{3} & +  0.02 x_{7}\\
 x_{2}   &  1.02444987775 & -0.14 x_{1} & +  1.66 x_{5} & -0.04 x_{13} & +  0.02 x_{8} & -0.57 x_{3} & -0.14 x_{7}\\
 x_{12}   &  41.1564792176 & +  1.48 x_{1} & -25.61 x_{5} & -0.07 x_{13} & -0.84 x_{8} & + 15.57 x_{3} & +  2.72 x_{7}\\
 x_{11}   &  6.81173594132 & -11.59 x_{1} & + 25.35 x_{5} & +  0.22 x_{13} & +  0.81 x_{8} & -25.23 x_{3} & -1.75 x_{7}\\
\hline
z    &  4.37897310513 & -2.24 x_{1} & +  0.16 x_{5} & -0.14 x_{13} & -0.12 x_{8} & -1.29 x_{3} & -0.12 x_{7}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  2.85379657287 & -0.37 x_{1} & -0.07 x_{9} & +  0.00 x_{13} & +  0.05 x_{8} & -0.61 x_{3} & +  0.01 x_{7}\\
 x_{4}   &  3.62537336897 & -0.72 x_{1} & +  0.04 x_{9} & +  0.09 x_{13} & -0.06 x_{8} & -1.27 x_{3} & -0.11 x_{7}\\
 x_{5}   &  1.20594246188 & +  0.41 x_{1} & -0.06 x_{9} & -0.04 x_{13} & -0.02 x_{8} & +  0.88 x_{3} & +  0.15 x_{7}\\
 x_{10}   &  14.7473667662 & +  2.05 x_{1} & -0.14 x_{9} & +  0.76 x_{13} & -0.00 x_{8} & -5.36 x_{3} & +  0.37 x_{7}\\
 x_{2}   &  3.02059424619 & +  0.54 x_{1} & -0.11 x_{9} & -0.10 x_{13} & -0.00 x_{8} & +  0.89 x_{3} & +  0.12 x_{7}\\
 x_{12}   &  10.2766860557 & -9.13 x_{1} & +  1.65 x_{9} & +  0.90 x_{13} & -0.44 x_{8} & -6.94 x_{3} & -1.18 x_{7}\\
 x_{11}   &  37.3878321019 & -1.09 x_{1} & -1.63 x_{9} & -0.73 x_{13} & +  0.41 x_{8} & -2.94 x_{3} & +  2.12 x_{7}\\
\hline
z    &  4.56767803804 & -2.17 x_{1} & -0.01 x_{9} & -0.15 x_{13} & -0.12 x_{8} & -1.15 x_{3} & -0.10 x_{7}\\
\end{array}\]
Final Dictionary
Solution:  4.56767803804
Num Pivots:  7
\end{document}
